The deficiency distance between a multinomial and a multivariate normal experiment is bounded under a condition that the parameters are bounded away from zero.

This result can be used as a key step in establishing asymptotic normal approximations to nonparametric density estimation experiments.

The bound relies on the recursive construction of explicit Markov kernels that can be used to reproduce one experiment from the other. The distance is then bounded using classic local-limit bounds between binomial and normal distributions. Some extensions to other appropriate normal experiments are also presented.